Illinois Journal of Mathematics

Fixed curves near fixed points

Alastair Fletcher

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Abstract

Let $H$ be a composition of an $\mathbb{R}$-linear planar mapping and $z\mapsto z^{n}$. We classify the dynamics of $H$ in terms of the parameters of the $\mathbb{R}$-linear mapping and the degree by associating a certain finite Blaschke product. We apply this classification to this situation where $z_{0}$ is a fixed point of a planar quasiregular mapping with constant complex dilatation in a neighbourhood of $z_{0}$. In particular, we find how many curves there are that are fixed by $f$ and that land at $z_{0}$.

Article information

Source
Illinois J. Math., Volume 59, Number 1 (2015), 189-217.

Dates
Received: 27 April 2015
Revised: 18 November 2015
First available in Project Euclid: 11 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1455203164

Digital Object Identifier
doi:10.1215/ijm/1455203164

Mathematical Reviews number (MathSciNet)
MR3459633

Zentralblatt MATH identifier
1337.30032

Subjects
Primary: 30C65: Quasiconformal mappings in $R^n$ , other generalizations
Secondary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX] 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04]

Citation

Fletcher, Alastair. Fixed curves near fixed points. Illinois J. Math. 59 (2015), no. 1, 189--217. doi:10.1215/ijm/1455203164. https://projecteuclid.org/euclid.ijm/1455203164


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